Operator mixing, UV asymptotics of nonplanar/planar $2$-point correlators, and nonperturbative large-$N$ expansion of QCD-like theories

TL;DR

This paper explores the relationship between perturbative operator mixing, UV asymptotics, and nonperturbative large-N expansion in QCD-like theories, providing explicit calculations and demonstrating universality in certain correlator ratios.

Contribution

It establishes a connection between perturbative ratios and nonperturbative large-N UV asymptotics, with explicit computations for scalar and twist-2 operators in YM theory.

Findings

Ratios of nonplanar to planar contributions match large-N UV asymptotics under certain conditions.

Universal value of ratio r_i = -1/N^2 for twist-2 operators.

Universality of r_i reflects the universality of the effective coupling in large-N YM.

Abstract

We work out the interplay between lowest-order perturbative computations in the 't Hooft coupling, $g^2=g^2_{YM} N$, operator mixing, renormalization-group (RG) improved ultraviolet (UV) asymptotics of leading-order (LO) nonplanar/planar contributions to $2$-point correlators, and nonperturbative large-$N$ expansion of perturbatively massless QCD-like theories. As concrete examples, we compute to the lowest perturbative order in $SU(N)$ YM theory the ratios, $r_i$, of LO-nonplanar to planar contributions to the $2$-point correlators in the orthogonal basis in the coordinate representation of the gauge-invariant dimension-$8$ scalar operators and all the twist-$2$ operators. We demonstrate that -- if $\frac{\gamma_0}{\beta_0}$ has no LO-nonplanar contribution, with $\gamma_0$ and $\beta_0$ the one-loop coefficients of the anomalous-dimension matrix and beta function respectively -- $r_i$ actually coincides with the corresponding ratio in the large-$N$ expansion of the RG-improved UV asymptotics of the $2$-point correlators, provided that a certain canonical nonresonant diagonal renormalization scheme exists for the corresponding operators. Contrary to the aforementioned scalar operators, for the first $10^3$ twist-$2$ operators we actually verify the above conditions, and we get the universal value $r_i=-\frac{1}{N^2}$. Hence, nonperturbatively such $r_i$ must coincide with the UV asymptotics of the ratio of the glueball self-energy loop to the glueball tree contribution to the $2$-point correlators above. As a consequence, the universality of $r_i$ reflects the universality of the effective coupling in the nonperturbative large-$N$ YM theory for the twist-$2$ operators in the coordinate representation.